Self-reference and Logic

Self-reference and Logic

Thomas Bolander

22nd August 2005

Self-reference is used to denote any situation in which someone or somethingrefers to itself. Object that refer to themselves are called self-referential. Anyobject that we can think of as referring to somethingor that has the abilityto refer to somethingis potentially self-referential. This covers objects such assentences, thoughts, computer programs, models, pictures, novels, etc.

The perhaps most famous case of self-reference is the one found in the Liarsentence:

This sentence is not true.

The Liar sentence is self-referential because of the occurrence of the indexicalthis sentence in the sentence. It is also paradoxical.1 That self-referencecan lead to paradoxes is the main reason why so much effort has been putinto understanding, modelling, and taming self-reference. If a theory allowsfor self-reference in one way or another it is likely to be inconsistent becauseself-reference allows us to construct paradoxes, i.e. contradictions, within thetheory. This applies, as we will see, to theories of sets in mathematics, theoriesof truth in the philosophy of language, and theories of introspection in artificialintelligence, amongst others.

This essay consists of two parts. The first is called Self-reference and thesecond is called Logic. In the first part we will try to give an account of thesituations in which self-reference is likely to occur. These can be divided intosituations involving reflection, situations involving universality, and situationsinvolving ungroundedness.2 In the second part we will turn to a more formaltreatment of self-reference, by formalizing a number of the situations involvingself-reference as theories of first-order predicate logic. It is shown that Tarskisschema T plays a central role in each of these formalizations.3 In particular,we show that each of the classical paradoxes of self-reference can be reduced to

1If the sentence is true, what it states must be the case. But it states that it itself is nottrue. Thus, if it is true, it is not true. On the contrary assumption, if the sentence is not true,then what it states must not be the case and, thus, it is true. Therefore, the sentence is trueiff it is not true.

2Often cases of self-reference will fit into more than one of these categories.3Tarskis schema T is the set of all first-order logical equivalences

T (

)

where is any sentence and

is a term denoting .

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schema T. This leads us to a discussion of schema T, the problems it gives riseto, and how to circumvent these problems.

The first part of the essay does not require any training in mathematicallogic.

Part I: Self-Reference

We start out by taking a closer look at paradoxes related to self-reference.

1 Paradoxes

A paradox is a seemingly sound piece of reasoning based on seemingly true as-sumptions, that leads to a contradiction (or other obviously false conclusion)(Audi, 1995). A classical example is Zenos Paradox of Achilles and theTortoise in which we seem to be able to prove that the tortoise can win anyrace against the much faster Achilles, if only the tortoise is given an arbitrarilysmall head-start (cf. (Erickson and Fossa, 1998) for a detailed description of thisparadox). Another classical paradox is the Liar Paradox, which is the contra-diction derived from the Liar sentence. Among the paradoxes we can distinguishthose which are related to self-reference. These are called the paradoxes ofself-reference. The Liar Paradox is one of these, and below we consider a fewof the others.

1.1 Grellings Paradox

A predicate is called heterological if it is not true of itself, that is, if it doesnot itself have the property that it expresses. Thus the predicate long isheterological, since it is not itself long (it consists only of four letters), but thepredicate short is not heterological. The question that leads to the paradoxis now:

Is heterological heterological?

It is easy to see that we run into a contradiction independently of whether weanswer yes or no to this question.

Grellings paradox is self-referential, since the definition of the predicateheterological refers to all predicates, including the predicate heterologicalitself.

1.2 Richards Paradox

Some phrases of the English language denote real numbers. For example, theratio between the circumference and diameter of a circle denotes the number pi.Assume that we have given an enumeration of all such phrases (e.g. by puttingthem into lexicographical order). Now consider the phrase

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the real number whose nth decimal place is 1 if the nth decimalplace of the nth phrase is 2, otherwise 1.

This phrase defines a real number, so it must be among the enumerated phrases,say number k in this enumeration. But, at the same time, by definition, it differsfrom the number denoted by the kth phrase in the kth decimal place.

Richards paradox is self-referential, since the defined phrase refers to allphrases that define real numbers, including itself.

1.3 Berrys Paradox

Berrys Paradox is obtained by considering the phrase

the least natural number not specifiable by a phrase containingfewer than 100 symbols.

The contradiction is that that natural number has just been specified using only87 symbols!

The paradoxes may seem simply like amusing quibbles. We may think ofthem as nothing more than this when they are part of our imprecise naturallanguage and not part of theories. When the reasoning and assumptions involvedin the paradoxes are not attempted to be made completely explicit and precise,we might expect contradictions to be derivable because of this lack of precision.But having a theorymathematical, philosophical or otherwisecontaining acontradiction is of course devastating for the theory. It shows the entire theoryto be inconsistent (unsound). The problem is that it turns out that in manyof the intuitively correct theories in which some kind of self-reference is takingplace, we can actually reconstruct the above paradoxes, and thereby show thesetheories to be inconsistent. This applies to the naive theories of truth, sets, andintrospection as we will later see.

Before we turn to a more thorough study of the situations in which self-reference is to be expected to occur, we put a bit more structure on our notionof self-reference by introducing reference relations.

2 Reference Relations

Reference can be thought of as a relation R between a class of referring objectsand a class of objects being referred to. R is called a reference relation, andit is characterized by the property that

(a, b) R iff b is referred to by a.

The domain of R, that is, the set of as for which there is a b with (a, b) R,is denoted dom(R). The range of R, that is, the set of bs for which there is ana with (a, b) R, is denoted ran(R). The relation R can be depicted as a graph

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small carcars

big car dom(R)

ran(R)

Figure 1: A reference relation.

T

truth is-relation

Figure 2: Reference relation for T .

on dom(R) ran(R), in which there is an edge from a dom(R) to b ran(R)iff (a, b) R. If e.g.

A = {small car, big car, cars}

and

B =

{,

}

we could have the reference relation depicted on Figure 1. If dom(R)ran(R) =, as above, a referring object will always be isolated from the object it refersto, since these two objects will be members of two distinct and disjoint classes.Self-reference is thus only possible when dom(R) ran(R) 6= .

Let T be the self-referential sentence

This sentence is true

(T for truth teller). The sentence refers to

(i) the sentence itself

(ii) the is-relation

(iii) the concept of truth.

Graphically, this could be represented by the reference relation in Figure 2.

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S2S1

is-relation

truth

dom(R)

ran(R)

Figure 3: Reference relation for S1 and S2.

Notice the loop at T . The loop means that

(T, T ) R,

that is, T is referred to by T , which is exactly the condition for T being self-referential. This leads us to the following definition:

An object a dom(R) is called directly self-referential ifthere is a loop at a in (the graph of) the reference relation.

Now consider the following two sentences, S1 and S2,

S1 : The sentence S2 is true.

S2 : The sentence S1 is true.

The reference relation for these two sentences become as depicted in Figure 3.Here the set of referring objects is dom(R) = {S1, S2} and the set of objectsreferred to is

ran(R) = {S1, S2, is-relation, truth} .

Notice that dom(R) ran(R) 6= . None of these sentences are directly self-referential, but S1 refers to S2 which in turn refers back to S1, and vice versa.This gives a cycle in the graph consisting of the nodes S1 and S2, and the twoedges connecting them. We consider both of S1 and S2 to be indirectly self-referential since each of them refers to itself through the other sentence. Thuswe define:

An object a dom(R) is called indirectly self-referential ifa is contained in a cycle in (the graph of) the reference relation.

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Kripke gives a very nice example of indirect self-reference in (Kripke, 1975). S1is the following statement, made by Jones,

S1 : Most of Nixons assertions about Watergate are false.

and S2 is the following statement, made by Nixon,

S2 : Everything Jones says about Watergate is true.

The reference relation for this pair of sentences will contain that of Figure 3,i.e. we have again a cycle between S1 and S2.

Let us consider a few additional examples of indirect self-reference. In thefollowing, when an object is either directly or indirectly self-referential, we oftensimply call it self-referential.

2.1 Naive Set Theory

In naive set theory (as conceived in the early works of Georg Cantor. See e.g.(Cantor, 1932)) the concept of a set can be defined in the following way:

By a set we understand any collection of mathematical objects (in-cluding sets).

We see that the concept of a set is defined in terms of mathematical objectswhich can themselves be sets. This means that what we have is a self-referentialdefinition of the concept of a set. This self-reference makes the defined conceptinconsistent, as we will see from Cantors Paradox, introduced in Section 4.

2.2 Dictionary Reference

In a dictionary, the referring objects are the definienda, that is, the expressionsor words being defined, and the objects referred to are the definientia, that is,the expressions or words that define the definienda. In Websters 1828 dictionarythe word regain is defined as:

regain : to recover, as what has escaped or been lost.

At the same time, the word recover is defined as:

recover : to regain; to get or obtain that which was lost.

Using only the words in italic, the reference relation for the above two dictionarydefinitions become as depicted in Figure 4. Since the definition of regain refersto the word recover and the definition of recover refers to the word regain,there is a cycle between these two words in the graph. Each is defined throughthe other in an indirectly self-referential way. This means that unless we knowthe meaning of one of these words in advance, the dictionary definition will notbe able to give us the full meaning of the other word.

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regain

recover

get

escaped

obtain

lost

Figure 4: A dictionary reference relation.

This becomes even worse if we consider an English dictionary of the entireEnglish language. Since every word is simply defined in terms of other words,we will not from the dictionary be able to learn the meaning of any of thewords, unless we know the meaning of some of them in advance. This makesa dictionary insufficient as a definition of meaning for a language, as noted byWittgenstein in the so-called Blue Book ((Wittgenstein, 1958)). Wittgensteinsway out was to think of a dictionary as supplied with a set of ostensive defini-tions. An ostensive definition of a word is a definition by pointing out thereferent of the worde.g. to say the word banjo while pointing to a banjo.

Wittgensteins ideas are related to ideas of groundedness of ungroundednessof reference relations, as we will see in Section 5. But before that we will relateself-reference to reflection and universality.

3 Reflection and Self-Reference

Self-reference is often an epiphenomenon of reflection of some kind. The wordreflection actually means bending back. We use reflection to denote situationssuch as: viewing yourself in a mirror; exercising introspection (that is, reflectingon yourself and your own thoughts and feelings); having a theory which iscontained in its own subject matter; having a picture which contains a pictureof itself (Figure 6). Reflection can also be considered as a name for all thesituations in which someone or something views itself from the outside. Inthe framework of reference relations, we can choose to define:

A reference relation R is said to have reflection if dom(R) ran(R).

By this definition, a reference relation has reflection iff every referring object isalso an object that is referred to. That is, if R is the reference relation of some

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Figure 5: Reflection means bending back.

Figure 6: A picture containing itself.

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Figure 7: An agent in Blocks World.

theory, then that theory can refer (represent, describe) not only objects of theexternal world but also all the objects of the theory itself.

Reflection does not in itself necessarily lead to self-reference, though self-reference often comes together with reflection. We do only have self-reference ifwe among the elements of dom(R) can point out an element r which refers tor. Reflection means that every element r of dom(R) is referred to by anotherelement q of dom(R), but for all such pairs (q, r) we might have q 6= r. InSection 4 we will show, though, that if reflection is combined with universality,then self-reference cannot be avoided.

Below we will consider some important examples of reflection.

3.1 Artificial Intelligence

A very explicit form of reflection is involved in the construction of artificialintelligence systems such as for instance robots. Such systems are called agents.Reflection enters the picture when we want to allow agents to reflect uponthemselves and their own thoughts, beliefs, and plans. Agents that have thisability we call introspective agents.

An artificial intelligence agent is most often equipped with some formallanguage which it uses for representing its experiences and beliefs, and which ituses for reasoning about its environment. That is, such an agent has a modelof the world it inhabits which is represented by a set of formal sentences.

Consider an agent situated in a blocks world 4 as depicted in Figure 7. Theagents task in this world is to move blocks to obtain some goal configuration(e.g. building a tower consisting of all blocks placed in a specific order). Theagents beliefs about this world could be represented in the agent by formal

4Blocks worlds are the classical example domains used in artificial intelligence.

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sentences such as

on(black box,floor)

on(dotted box, black box)

on(white box,floor)

on(agent,floor).

For the agent to be introspective, though, it should also contain sentences con-cerning the agents own beliefs. If the agent believes the sentence

on(black box,floor)

to be part of its own model of the world, that could e.g. be represented by thesentence

agent(pon(black box,floor)q).

Now, the referring objects in this situation are obviously the sentences thatmake up the agents model of the world. So if R denotes the reference relation ofthe agent, then dom(R) consists of all these sentences. The object referred to inthe case of a sentence like on(black box,floor) is the black box on the floor, whilethe object referred to in the case of a sentence like agent(pon(black box,floor)q)is the sentence on(black box,floor). If is any sentence then agent(pq) is asentence referring to . This means that the set of objects referred to, ran(R),contains every sentence, i.e. we have dom(R) ran(R). By our definition, thismeans that R has reflection. This reflectionthat the agent can refer to any ofits own referring objectsturns out to provide a major theoretical obstacle tothe construction of introspective agents, as we will see in Section 11.

3.2 Philosophy of Language

One of the major problems in the philosophy of language is to give a definitionof truth for natural languages. Tarski suggests that every adequate theory oftruth should give a predicate true satisfying

is true iff

where is any sentence. In such a theory of truth we would also have reflection,since the referring objects are sentences, and any sentence can be referred toby the sentence

is true.

Reflection is in itself not enough to give self-reference. In both examplesabove we had reflection but no self-reference, since there were no cycles in thereference relations. The problem is, though, that reflection often comes togetherwith universality, and when we have both reflection and universality then self-reference cannot be avoided. Universality is the subject of the following section.

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4 Universality and Self-Reference

When we make a statement about all entities in the world, this will necessarilyalso cover the statement itself. Thus such statements will necessarily be self-referential. We call such statements universal (as we call formulas of the formx(x) in predicate logic). Actually, we will use the term universal to de-note any statement concerning all entities in the relevant domain of discourse.Correspondingly, in the framework of a reference relation R, we can define:

An object a dom(R) is called universal if (a, b) R for allb ran(R).

If R is the reference relation of our natural language then the sentence

All sentences are false (1)

will be universal. The problem about universality is that reflection and uni-versality together necessarily lead to self-reference, and thereby is likely to giverise to paradoxes. To see that reflection and universality together lead to self-reference, assume R has reflection and that a dom(R) is a universal object.Then we have (a, b) R for all b ran(R), and since dom(R) ran(R) weespecially get (a, a) R. That is, we have the following result:

Assume R has reflection and that a dom(R) is universal. Thena is self-referential.

Universality enters the picture in the two examples of reflection previously givenif we want the agent to be able to express universal statements about its envi-ronment or if we want to be able to apply the truth predicate to sentences thatconcern all sentences of the language (like e.g. the sentence (1)). In such casesself-reference cannot be avoided, and as we will see in the second part of theessay this will allow the paradoxes to surface and produce contradictions in theinvolved theories.

The problem sketched is not in any way only related to theories of agentintrospection and truth. Any theory that is part of its own subject matterhas reflection. Thus, if these theories make use of universal statements as well,then these theories contain self-referential statements, and then the paradoxes ofself-reference will not be far away. Thus, self-reference is a problem to be takenseriously by any theory that is part of its own subject matter. This appliesto theories of cognitive science, psychology, semiotics, mathematics, sociology,system science, cybernetics, computer science.

Note, that each of the paradoxes of self-reference considered in Section 1involves both reflection and universality, since they all refer to the totality ofobjects of their own type: the predicate heterological refers to all predicates;the phrase defining a real number in Richards paradox refers to all phrasesdefining real numbers; the phrase specifying a natural number in Berrys para-dox refers to all phrases specifying natural numbers.

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Let us conclude this section by considering another example of a universalobject in a reflective setting. In the naive theory of sets (cf. Section 2.1) we canconsider the set U of all sets. U is certainly a universal object, since it refersto all other sets.5 At the same time, the theory of sets is reflective since forthe reference relation R of sets, dom(R) and ran(R) are both the class of allsets. Thus U is a self-referential object, and this leads to trouble. Cantor haveproved that the cardinality6 of any set is smaller than the cardinality of the setof subsets of this set. This result is called Cantors Theorem.7 Let us seewhat happens if we apply Cantors Theorem to the set U . First of all, we notethat the set of all subsets of U is U itself, since U contains all sets. But then,by Cantors Theorem, the cardinality of U is smaller than the cardinality of U ,which is a contradiction. This contradiction is known as Cantors Paradox.Cantors Paradox proves that the naive theory of sets is inconsistent.

5 Ungroundedness and Self-Reference

Self-reference often occurs in situations that have an ungrounded nature. Givena reference relation R, we can define ungroundedness in the following way:

An object a dom(R) is called ungrounded if there is aninfinite path starting at a in the graph of the reference relationR. Otherwise a is called grounded.

Note, that if dom(R)ran(R) = , that is, if referring objects and objects beingreferred to are completely separated, then all elements are grounded.

If we take the dictionary example of Section 2.2, we can give a simple exampleof ungroundedness. Let R be the reference relation of Websters 1828 dictionary,that is, let R contain all pairs (a, b) for which b is a word occurring in thedefinition of a. Since every word of the dictionary refers to at least one otherword, every word will be the starting word of an infinite path of R. Here is afinite segment of one of these paths, taken from the 1828 dictionary:

regain recover lost mislaid laid

position placed fixed . . .

Now the problem that Wittgenstein considered can be stated in the followingsimple manner: in a dictionary all words are ungrounded. Since there are onlyfinitely many words in the English language, any infinite path of words willcontain repetitions. If a word occurs at least twice on the same path, it will

5It is natural to think of the objects being referred to by a set as the elements of the set.6The cardinality of a set is a measure of its size.7It is interesting at this point to note that the argument leading to Cantors Theorema

so-called diagonal argument (which he was the first to use)has basically the same structureas Richards Paradox.

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be contained in a cycle. Thus, in any dictionary of the entire English languagethere will necessarily be words defined indirectly in terms of themselves. Thatis, any such dictionary will contain (indirect) self-reference.

Ungroundedness does not always lead to self-reference, but self-reference isvery often a byproduct of ungroundedness. So whenever one encounters un-groundedness, one should be very careful to ensure that this ungroundednessdoes not lead to self-reference and paradoxes.

Actually, as showed by Steven Yablo in (Yablo, 1993), ungroundedness canlead to paradoxes even in cases where we do not have self-reference. YablosParadox is obtained by considering an infinite sequence of sentences S1, S2, . . .defined by:

S1 : All sentences Si with i > 1 are false.



...

The reference relation for these sentences looks like this:

S1 //)) )) ))

S2 // 55 77S3 // 55S4 // S5 //

As one sees, there is no self-reference involved, but we still get a paradox:Assume Si is true for some i. Then all Sj for j > i must be false. In particular,Si+1 must be false. But since Sj is false for all j > i+1, Si+1 must also be true.This is a contradiction. Therefore all Si must be false. But then S1 should betrue, which is again a contradiction.

It should be noted that even though ungroundedness does not always lead toself-reference, self-reference always leads to ungroundedness: any self-referentialobject a is contained in a cycle, and we get an infinite path from a by passingthrough this cycle repeatedly.

6 Vicious and Innocuous Self-Reference

Not all self-reference leads to paradoxes. There is no paradox involved in aself-referential sentence like

This sentence is true. (2)

We can assume either that the sentence is true or that it is false, and neither ofthe cases will lead into contradiction. But as soon as we introduce a not inthe sentence, that is, consider the following sentence instead

This sentence is not true (3)

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we get a paradox (the Liar Paradox). Self-reference that leads to paradoxes wecall vicious self-reference and self-reference that does not we call innocuousself-reference. It can be shown that self-reference can only be vicious if itinvolves negation or something equivalent (as the not in (3)). This means, forinstance, that none of the paradoxes of self-reference considered above couldbe carried through if the occurrence of negation in their central definitionswhere removed (e.g. if we removed the not in the definition of heterologicalof Grellings Paradox).

Part II: Logic

We now turn to a more formal treatment of self-reference, by formalizing someof the situations considered in the first part of the essay as theories of first-orderpredicate logic (henceforth simply called first-order theories). We will assumethat all considered first-order theories contain the standard numerals:8

0, 1, 2, 3, . . .

We use pq to range over coding schemes. By a coding scheme we understandany injective mapping from sentences into numerals. That is, if is a sentencethen pq is the numeral n for some natural number n. pq is a name for ;we call it the code number of . If (x) is a formula containing x as itsonly free variable then (pq) is a sentence expressing that has the propertyexpressed by . In this sense, (pq) refers to .

Schema T is, as before, defined as the theory containing each of the equiv-alences

T (pq)

where T is a fixed one-place predicate symbol and is any sentence.The aim of this part of the essay is to show that schema T is taking a central

position in almost all situations in which we have self-reference. Indeed, schemaT can be thought of as a unifying principle of all the different occurrences ofself-reference.

7 Formalizing Paradoxes

We now try to formalize some of the most famous paradoxes of self-referenceto show how these involve schema T. As mentioned, a paradox is a seeminglysound piece of reasoning based on seemingly true assumptions that lead to acontradiction. Formalizing a paradox means to reconstruct it inside a formaltheory (in our case, a first-order theory). This involves finding formal coun-terparts to each of the elements involved in the informal paradox. The formalcounterpart of a piece of reasoning is a formal proof and the formal counter-part of an assumption is an axiom. Thus the formal counterpart of a piece ofreasoning leading to a contradiction will be a formal proof of the inconsistencyof the theory in question. Thus:

8Actually, any infinite collection of closed terms would do.

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A formalization of a paradox is a formal proof of the incon-sistency of the theory in which the axioms are the formal coun-terparts of the assumptions of the paradox.

7.1 The Liar Paradox

As already mentioned, the Liar Paradox is the contradiction that emerges fromtrying to determine whether the Liar sentence

This sentence is false

is true or false. We will now try to formalize this paradox.In general, sentences that are directly self-referential can be put in the fol-

lowing form:This sentence has property P. (4)

The assumption that characterizes such a sentence is that the term this sen-tence refers to the sentence itself. Another way of stating this assumption isto say that (4) should satisfy the following equivalence

This sentence has property P

This sentence has property P has property P ,(5)

that is, replacing the term this sentence by the sentence itself will not changethe meaning of the sentence. Formally, this assumption can be expressed as theaxiom

P (t) P (pP (t)q) (6)

where t is a term having the intended interpretation: this sentence. Thisequivalence corresponds to the equivalence (5), in that this sentence havebeen replaced by t and the quotes have been replaced by pq.

The Liar Paradox also rests on the assumption that our language has a truthpredicate. The formal counterpart of this assumption is that our theory includesschema T. In the Liar sentence, P is the property not true. Let therefore Pin (6) denote the formula T (x). Then, in the theory consisting of schema Tand (6), we get the following proof:

1. T (t) T (pT (t)q) (6) with P being T2. T (pT (t)q) T (t) instance of schema T3. T (pT (t)q) T (pT (t)q) by 1. and 2.

This proves the theory consisting of (6) and schema T to be inconsistent, whichis our formalization of the Liar Paradox.

7.2 Grellings Paradox

We will now formalize Grellings paradox. Recall that Grellings Paradox isthe paradox that emerges when trying to answer whether heterological is

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heterological. The formal counterpart of a predicate is a formula. A formula(x) is then heterological if it is not true of itself, that is, if

T (p(pq)q)

holds, where T is a truth predicate. So to formalize Grellings paradox we againneed to have schema T among our axioms. We also need axioms that allowus to apply a formula to itself (that is, the code of itself). To obtain this, weintroduce a function symbol app and axioms

app (p(x1)q, ) = p()q (7)

for all formulas and all terms . These axioms ensure us that app(p(x1)q, )denotes the result of applying (x1) to (that is, instantiating (x1) with). Now we can formalize the predicate heterological as the formula het(x1)given by

het(x1) =df T (app(x1, x1)) .

To obtain the contradiction we should ask whether het(phet(x1)q) holds or not.We get the following proof:

1. het(phet(x1)q) T (app (phet(x1)q, phet(x1)q)) by def. of het(x1)2. het(phet(x1)q) T (phet(phet(x1)q)q) by 1. and (7)3. het(phet(x1)q) T (phet(phet(x1)q)q) instance of schema T4. T (phet(phet(x1)q)q) T (phet(phet(x1)q)q) by 2. and 3.

This proves the theory consisting of (7) and schema T to be inconsistent, whichis our formalization of Grellings paradox.

Richards Paradox is formalized in much the same way as Grellings Paradox,though the formalization becomes slightly more technical. For these reasons wechoose to leave out a formalization of Richards Paradox in this essay.

7.3 Berrys Paradox

Obviously, to formalize Berrys Paradox, we need axioms formalizing a reason-able part of arithmetic. Apart from this we only need a formal counterpart ofthe notion of specifiability (the formal counterpart of a phrase naturally beinga formula). We can use the same trick as we did in the previous examples. Aformula (x) specifies the number n iff (m) holds exactly when m = n. Ifwe want to define a formula spec(x, y) such that spec(p(x)q, n) holds preciselywhen (x) specifies n, then it should look like

spec(x, y) =df z (z = y T (app(x, z)))

where T and app are defined as before. We will not go further into the details offormalizing this paradox, but refer to (Boolos, 1989) in which this is carried out.We just note that again schema T is central to the formalization. The notion

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of specifiability could not have been formalized without schema T or somethingequivalent.

We have now shown how to formalize several of the most famous paradoxesof self-reference, and, as we have seen, these formalized paradoxes all turn outto be reducible to schema T. That is, all these paradoxes have a common corewhich is schema T. What we can conclude is that:

(i) That schema T can be extracted from all these paradoxes helps us see theclose formal relationship between the paradoxes of self-reference.

(ii) That all these paradoxes can be extracted from schema T helps us tosee the importance of schema T in understanding the paradoxes of self-reference, and in understanding self-reference in general.

Below we consider some examples of occurrences of schema T in the philos-ophy of language, mathematics, and artificial intelligence.

8 The Naive Theory of Truth

As mentioned, Tarski thought of his schema T as describing the principle thatany theory of truth should satisfy. The first-order theory consisting only ofschema T is consistent. But for schema T to be a sensible principle of truth wemust expect it to be consistent also when added to any consistent, realisticfirst-order theory. It should be a principle of truth working no matter whichdomain of discourse we would like to apply truth to. But, unfortunately, becauseof self-reference it is not so. In the formalizations of the paradoxes above we haveseen several examples showing that schema T becomes inconsistent when addedto even quite weak and harmless axioms (at least harmless when these axiomsare taken by themselves or together with standard theories for arithmetic, settheory, or the like). In fact, it can easily be shown that all of the axioms assumedabove in addition to schema T are interpretable in Peano Arithmetic, that is,they can all be translated into equivalent axioms of Peano Arithmetic.9 Thisgives us the famous Tarskis Theorem:

Peano Arithmetic extended with schema T is inconsistent.

Note the interesting fact that any of the above paradoxes can be used to proveTarskis Theoremone just needs to show that the axioms of the formalizedparadox are interpretable in PA (Peano Arithmetic). This shows that thecontradiction derivable from the formalized paradox can be carried throughin PA + schema T.

9For a precise definition of interpretable in we refer to (Mendelson, 1997) or a similarintroduction to mathematical logic. At this point it is enough to note that when an axiomA is interpretable in a theory K it means that any proof in K + A can be translated into acorresponding proof in K. It should be noted that to prove the interpretability we need tochoose our coding scheme

with care.

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That schema T becomes inconsistent when standard arithmetic is added isa very serious drawback for the theory of truth expressed through schema T. Itgives rise to an important problem of how we can restrict schema T to regainthe essential consistency. This is the question that we take up in Section 12.

But let us first consider some more examples of situations in which schemaT turns up, which makes the reasons to find consistent ways to restrict schemaT even more urgent.

9 Godels Incompleteness Theorem

We now show how schema T is related to Godels famous First IncompletenessTheorem.

A version of the Incompleteness Theorem states that

If PA is -consistent10 then it is incomplete.11

To prove this, we can show that the assumption that PA is both -consistentand complete leads to a contradiction. On the basis of the formalizations ofparadoxes that we have been considering, we see that this could be proved byshowing that if PA were both -consistent and complete then some paradoxwould be formalizable in PA. This was, roughly, Godels idea.12 He constructeda formula Bew (for Beweis) in his theory satisfying, for all and all n,

` Bew(n, pq) n denotes a proof of . (8)

Assuming the theory to be -consistent and complete we can prove that

` xBew(x, pq) `

for every sentence . The proof runs like this: First we prove the implica-tion from left to right. If ` xBew(x, pq) then there is some n such that6` Bew(n, pq), by -consistency. By completeness we get ` Bew(n, pq) forthis n. By (8) above we get that n denotes a proof of . That is, is provable,so we have ` . To prove the implication from right to left, note that if ` then there must be an n such that ` Bew(n, pq), by (8). From this we get` xBew(x, pq), as required. This concludes the proof.

Now, when we have

` xBew(x, pq) `

in a complete theory, we must also have

` xBew(x, pq) .

10A theory is called -consistent if, for every formula (x) containing x as its only freevariable, if ` (n) for every natural number number n, then it is not the case that ` x(x).

11A theory is incomplete if it contains a formula which can neither be proved nor disproved.12Though he considered a different formal theory, P.

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If we let the formula xBew(x, pq) be abbreviated by T (pq) then these equiv-alences read

` T (pq)

which is schema T!That is, if we assume PA (or a related theory) to be -consistent and com-

plete then schema T turns out to be interpretable in it. Now, Tarskis Theoremshows that there exists no such consistent theory. This gives us a proof ofGodels Incompleteness Theorem. Furthermore, in the same way that one coulduse any of the paradoxes of self-reference to prove Tarskis Theorem, one canuse ones favorite paradox of self-reference to prove Godels Theorem.

To summarize the process: first you assume your theory to be both -consistent and complete. Then you show that this makes schema T interpretablein the theory. Having schema T means that you can choose any paradox of self-reference and formalize it in the theory. The formalized paradox produces acontradiction in the theory, and thus shows that the theory cannot be both-consistent and complete.

Godel himself actually had a footnote in his 1931 article, in which he provedthe Incompleteness Theorem ((Godel, 1931)), saying that any paradox of self-reference13 could be used to prove the Incompleteness Theorem.

The reason that we have a result such as Godels Incompleteness Theorem isclosely related to reflection. What Godel ingeniously discovered was that formaltheories can be reflected inside themselves, since numerals can be used to referto formulas through the use of a coding scheme, pq, and by means of thesecodes provability can be restated inside the theories as arithmetical properties.

10 Axiomatic Set Theory

Schema T also plays a central role in axiomatic set theory. By the full ab-straction principle we understand the set of formulas on the form

x (x {y | (y)} (x)) 14

where is any formula. When Gottlob Frege tried to give a foundation formathematics (set theory) through his works Die Grundlagen der Arithmetik(1884) and Grundgesetze der Arithmetik (1893,1903), the full abstractionprinciple were among his axioms. But in 1902 his system was shown to beinconsistent by Bertrand Russell. Russell constructed a paradox of self-referencewhich was formalizable within Freges system. Russells Paradox runs likethis:

Let M be the set of all sets that are not members of themselves. IsM a member of itself or not?

13He used the term epistemic about these paradoxes.14The formula can be read: for all sets x, x is in the set of ys for which (y) holds if and

and only if (x) holds.

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From each answer to this question the opposite follows. Notice the similaritybetween this paradox and Grellings paradox considered in Section 1.1. Rus-sells Paradox can be formalized in any system containing the full abstractionprinciple. We let M = {y | y 6 y}, that is, M = {y | (y)} where (y) = y 6 y.The abstraction principle instantiated by the formula now becomes

x (x {y | y 6 y} x 6 x) .

Letting x = {y | y 6 y}, we get

{y | y 6 y} {y | y 6 y} {y | y 6 y} 6 {y | y 6 y}

which is a contradiction. Thus Freges system, or indeed any system containingthe full abstraction principle, is inconsistent.

The discovery of this inconsistency lead to extensive research in how the fullabstraction principle could be restricted to regain consistency.

Actually, as we will now show, every instance of schema T can be interpretedin the corresponding instance of the abstraction principle. This means that if wecan prove that a set of instances of schema T is inconsistent, then we have alsoproven that the corresponding set of instances of the abstraction principle is in-consistent. In other words, proving consistency results about restricted versionsof schema T will also give corresponding consistency results about restrictedversions of the abstraction principle.

The result is the following:

Every instance of schema T:

T (pq)

can be interpreted in the corresponding instance of the abstrac-tion principle:

x (x {y | } ) .

The proof is quite simple. If we have got a theory containing

x (x {y | } )

then T (pq) can be interpreted in it by the following extension by definitions :15

pq = {y | }

T (x) 0 x.

Since we havex (x {y | } )

15We refer again to (Mendelson, 1997) for a definition of the concept of extension bydefinitions.

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we get in particular(0 {y | } )

which is the same asT (pq) ,

using the definitions of pq and T . This proves T (pq) to be interpretablein x (x {y | } ).

11 Agent Introspection

We now turn to our last example of an occurrence of schema T in a situationdealing with self-reference. We consider again the problem of constructing in-trospective agents, as introduced in Section 3.1. Since the agents model of theworld is supposed to consist of a set of sentences, we can think of this model asbeing a formal theory K. This could be a theory in any kind of formal language,but at this point we will assume that it is a theory in a first-order language.Then, for the agent to believe that e.g. the black box is on the floor wouldcorrespond to having

K ` on(black box,floor). (9)

If the agent has introspection, it also has beliefs about its own model of theworld. If it believes the sentence in (9) to be contained in its own model of theworld we would have

K ` agent (pon(black box,floor)q) .

Now, if we assume that all of the agents beliefs about itself to be correct, weshould have

K ` agent(pq) K `

for all sentences . Of course, not all of an agents beliefs about itself willnecessarily always be correct. But even so, the agent might believe this to bethe case; and that would correspond to having

K ` agent(pq)

for all sentences . Using T instead of agent this gives us, once again,schema T!

That is, if an agent has introspection and believes this introspection to becorrect, then it will necessarily contain schema T in its model of the world.As we know from Tarskis Theorem and our formalized paradoxes this is verydifficult to obtain without running into contradictions. At least, it is extremelysensitive to what other axioms we have in K. This is a major drawback in thedesign of introspective agents.

We have to expect that any kind of axioms could be in K, depending on theenvironment of the agent and its beliefs about it. The set of axioms of K couldeven change over time due to changes in the environment. If K includes schema

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T it means that the agent could suddenly become inconsistent as a consequenceof changes in the external world. This seems to prove that it is not possiblefor an introspective agent consistently to obtain and retain the belief that itsintrospection is correct.

This conclusion appears very counterintuitive, but again it has to do withthe paradoxes of self-reference. If the agent has introspection, and believesthis introspection to be correct, it can construct paradoxes of self-referenceconcerning its own beliefs, and these paradoxes make the agent inconsistent.

The problem is now to find ways to treat agent introspection such thatthis introspection will not lead into inconsistency. It seems that we have twopossibilities: either to ensure that the agent will not be able to make self-referential statements (which would be a restriction on its introspective abilities)or to restrict its logical abilities such that self-referential statements could beassumed consistently. Such restrictions are the subject of the following section.

12 Taming Self-Reference

We have now seen that schema T occurs as the natural principle in a large num-ber of situations of very different kinds. Schema T is the underlying principlein the naive theories of truth, sets, and agent introspection. But unfortunately,schema T is also the underlying principle in the paradoxes of self-reference,which means that most of the theories we are interested in become inconsistentwhen schema T is added. Since the inconsistency of schema T is a consequenceof the presence of self-referential sentences, there seems to be two possible waysto get rid of the problem: ban self-referential sentences in our language orweaken the underlying logic so that these sentences will do no harm. That is,the different ways to restrict schema T in order to ensure consistency seems todivide into the following two major categories:

(i) Cutting away the problematic part (i.e. getting rid of the viciously self-referential sentences).

(ii) Making the problematic part unproblematic (i.e. ensure that self-referencedoes not lead to disaster).

12.1 Cutting Away the Problematic Part

Cutting away the problematic part means to restrict the set of instances ofschema T such that the viciously self-referential sentences are excluded fromentering the schema. By the T-scheme over M , where M is a set of sentences,we understand the following set of equivalences:

T (pq) , for all M.

If M does not contain sentences that are viciously self-referential, it can beproven that the T-scheme over M can consistently be added to any consistent

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theory.16 This is because banning the viciously self-referential sentences fromschema T makes it impossible to reconstruct the paradoxes of self-referencewithin the theory.

One very coarse way of disallowing self-reference was proposed by Tarskihimself ((Tarski, 1956)): M should not be allowed to contain any sentence inwhich the predicate symbol T occurs. Note that this will ensure that none ofthe proofs of the formalized paradoxes considered in Section 7 can be carriedthrough. This restriction is sufficient to reestablish consistency, but it is at theexpense of a substantial loss of the expressive power of schema T. It means thatiterated truth like in

It is true that it is not true that n is a prime number

that formally looks like this

T (pT (prime(n))q)

will not be treated correctly by the restricted T-scheme.Several less coarse solutions have been proposed in the literature since Tarski.

First of all, one notes that not all self-reference is vicious, so we can allow self-referential sentences in M as long as they are not vicious. As mentioned, forself-reference to be vicious, it needs to involve negation. A sentence in whichthe predicate symbol T is not in the scope of negation () is called a positivesentence. Positive sentences can be self-referential, but only of the innocuouskind. Donald Perlis and Solomon Fefermann showed independently ((Perlis,1985), (Feferman, 1984)) that the T-schema over a set of positive sentences canconsistently be added to any consistent theory.

Another way to exclude viciously self-referential sentences is to make restric-tions on universality. As we saw in Section 4, reflection only necessarily leadsto self-reference when it is combined with universality. Refraining from havinguniversal sentences about truth like e.g.

All sentences are true

in M we can again obtain a consistent, restricted T-scheme. More precisely, in(Bolander, 2002) it is shown that if none of the sentences of M contain T (x) asa sub-formula with x quantified, then the T-scheme over M can consistently beadded to any consistent theory.

Finally, the method of restricting negation and the method of restrictinguniversality can be combined to get an even stronger T-scheme. M can consis-tently be allowed to contain any sentence in which T (x) does not occur in thescope of negation (see (Bolander, 2002)).

16That is, can consistently be added to any consistent theory that does not in advancecontain axioms for the T predicate.

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12.2 Making the Problematic Part Unproblematic

Another way of ensuring consistency is to stick with self-reference (i.e. all in-stances of schema T) but to make sure that self-reference does not get the chanceto become paradoxical. Such solutions seem again to divide into two categories:

(i) Restricting the form of schema T.

(ii) Restricting the underlying logic.

Below we consider each of these methods.

Restricting the Form of Schema T

Instead of having bi-implications

T (pq) (10)

in some cases it is sufficient to have e.g. the following implications

T (pq) and T (pq) T (pT (pq)q).

Some of these restrictions on the form of schema T will form consistent exten-sions to any consistent theory, even if we do not restrict the set of sentencesthat these schemas are instantiated with. Results of this type can be found ine.g. (Montague, 1963), (Thomason, 1980), and (McGee, 1985). Another pos-sibility is to use a weak equivalence operator in (10) instead of the classicalbi-implication operator . A result concerning such a weak equivalence opera-tor can be found in (Feferman, 1984).

Restricting the underlying logic

Theories containing schema T become inconsistent because in them we canconstruct self-referential sentences that turn out to be true iff they are false.If we change the underlying logic such that sentences are allowed either to benor true nor false, or both true and false, the self-referential sentences willno longer be able to prove the theories to be inconsistent. Kripke considersin (Kripke, 1975) the possibility of allowing sentences to have no truth-value,that is, to be neither true nor false. His trick is then to only assign truth-values to the grounded sentences of the language (cf. Section 5). By this, heensures that no self-referential sentence will be given a truth-value (since everyself-referential sentence is ungrounded). This corresponds to the fact that in adictionary, as considered in Section 5, we can only, from the dictionary alone,assign meaning to the grounded words. The ungrounded words, among thesethe self-referentially defined ones, will be undecided (not be assigned anymeaning). Kripkes theory can be used to construct formal systems in which weconsistently have schema T, but in which the underlying logic is restricted (wecannot have classical negation, for instance, since this requires every sentence tobe either true or false). Graham Priest (in (Priest, 1989) and others) proposesthat we should allow sentences to be both true and false, because this is, in asense, what paradoxical self-referential sentences are.

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13 Conclusion

The paradoxes of self-reference still have no final solution that is generally agreedupon. This makes them, in a sense, genuine paradoxes. The presence of aparadox is always a symptom that some part of our fundamental understandingof a subject is crucially flawed. In Zenos Paradox it was the understandingof infinity that was deficient. In the paradoxes of self-reference it seems thatwhat we do not yet have a proper understanding of is the fundamental relationbetween something that refers (or represents) and something that is referredto (or represented) when these two can not be completely separated. As longas this relationship is not entirely grasped we will probably not get to a fullunderstanding of the paradoxes of self-reference and their consequences for thetheories of truth, sets, agent introspection, etc.

In Zenos Paradox it was not an explicitly stated assumption that laterproved to be defective. In the paradox it was implicitly assumed that infinitelymany things can not happen in finite time, but it was not until the developmentof the mathematical calculus that this assumption could be made explicit andrejected. In the case of Zenos Paradox it was thus not simply a question offinding the failing assumption involved in the paradox. It was rather a questionof discovering a new dimension of the world that had hitherto been hidden tothe human eye. A similar thing might very well be the case for the paradoxes ofself-reference. The right solution (assuming there is one) to the paradoxes is notto remove or restrict any of our explicit assumptions (that is, restrict schemaT, underlying logic, or similar), but to discover a new dimension of the problemthat will in the end give more, not fewer, axioms in some kind of extendedlogic. This new dimension is then expected to make explicit some assumptionsabout the general relations between referring objects and objects referred to;assumptions that a now invisible to us.17

Finally: What would be a suitable concluding remark in an essay like this?18

References

Audi, R., editor (1995). The Cambridge Dictionary of Philosophy. CambridgeUniversity Press.

Bartlett, S. J., editor (1992). ReflexivityA Source-Book in Self-Reference.North-Holland, Amsterdam.

Bolander, T. (2002). Restricted truth predicates in first-order logic. (submittedfor publication). To be represented at LOGICA 2002.

17It has been proposed to solve the paradoxes of self-reference by extending logic to includecontexts. Then a paradox such as the Liar will be resolved by the fact that the context beforethe Liar sentence is uttered is different from the context after it has been uttered. Such asolution seems to be of the kind I propose here, since the logic is extended (with contexts)rather than restricted. But, it should be noted that if we are allowed freely to refer to contextsthen the Liar sentence can be strengthened to give a paradox even in this theory.

18Answer: A self-referential question which is its own answer.

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Perlis, D. and Subrahmanian, V. S. (1994). Meta-languages, reflection principlesand self-reference. In Handbook of Logic in Artificial Intelligence and LogicProgramming, volume 2, pages 323358. Oxford University Press.

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Self-reference and Logic